Minimizing Movements for Mean Curvature Flow of Partitions

Abstract

Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global 1 n+1 -Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to 1 2 in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.

Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

“…In this section we prove density estimates for almost minimizers (see theorem 3.2). In the two-phase case without mobility, density estimates have been proven in [1,42] (see also the proof of theorem 4.2 for the case with mobility and forcing) and in the isotropic N -phase case is proven in [14]. The proof of theorem 3.2 is similar to [14, theorem 3.6], however, some (technical) difficulties arise when two anisotropies differ too much and this is why we need assumption (3.5) for proving the lower-density estimates.…”

“…The following volume-distance comparison appeared in a similar form also in [1,14,42] and will be used in the proof of the existence of GMM.…”

“…This has been done in several different ways: just to quote a few, the Brakke varifold-solution [15], the viscosity solution (see [30] and references therein), the Ilmanen elliptic regularization [37], the level-set theoretic subsolution and the minimal barrier solution (see [12] and references therein), the Almgren-Taylor-Wang [1] and Luckhaus-Sturzenhecker [42] solutions, next included by De Giorgi into his notion of minimizing movement and generalized minimizing movement (GMM) [21,22]; see also [16,24,47]. Some of those solutions (e.g., the Brakke solution [15,54], the GMM solution [14], the elliptic regularization [50]) can be adapted to treat the multiphase case at least in the Euclidean case, especially those that do not rely heavily on the comparison principle. Also, the existence of a distributional solution of mean curvature evolution of partitions on the torus using the time thresholding method introduced in [46] has been proved in [41]; see also [39].…”

Brakke’s inequality for the thresholding scheme

“…Chambolle [9] showed that the scheme [2,25] which seems academic at a first glance can be implemented rather efficiently. Recently, Bellettini and Kholmatov [6] analyzed the scheme in the multi-phase case. However, neither a conditional convergence result to a distributional BV-solution, nor one to a Brakke flow are available yet.…”

A Second Order Gradient Flow of p-Elastic Planar Networks

“…Such time-discrete schemes have been used in the study of geometric evolutions starting from the pivotal works by Almgren, Taylor and Wang [2] and by Luckhaus and Sturzenhecker [23] in the case of the mean curvature flow. An extension of these techniques to multiple-phase systems can be found in [4], while an adaptation to the L 2 -gradient flow for the elastic energy (p = 2) of an open curve, which gives rise to a fourth order flow, has been recently proposed in [13,3].…”